Heat olive oil in a large, deep skillet over medium heat. Add onion with a pinch of salt; cook and stir until onion is translucent, about 5 minutes. Stir in sausage and cook until browned, 5 to 7 minutes.

Pour 1 1/2 cups chicken broth into sausage mixture and bring to a boil while scraping the browned bits off the pan with a wooden spoon.

Add orecchiette pasta; cook and stir pasta in broth, adding remaining broth when liquid is absorbed, until pasta is cooked through, about 15 minutes.

Stir arugula into sausage-pasta mixture until arugula wilts.

Ladle pasta into bowls and sprinkle with Parmigiano-Reggiano cheese.